Sciresol Sciresol https://indjst.org/author-guidelines Indian Journal of Science and Technology 0974-5645 10.17485/IJST/v15i46.1862 research article Lucas Antimagic Labeling of some Star Related Graphs Sumathi P 1 Chandravadana N chandraresearch21@gmail.com 2 Head & Associate Professor, Department of Mathematics, C. Kandaswami Naidu College for Men Chennai, Tamil Nadu India Assistant Professor, Department of Mathematics, Justice Basheer Ahmed Sayeed College for Women, Teynampet Chennai, Tamil Nadu India 10 12 2022 15 46 2542 15 9 2022 25 9 2022 2022 Abstract

Objective: To identify a new family of Lucas antimagic graph. Methods: A (p,q) graph G is said to be a Lucas antimagic graph if there exists a bijection f: E(G){L1,L2,Lq} such that the induced injective function f* : V(G){1,2,Lq} given by f*u=eEufe are all distinct (where E(u) is the set of edges incident to u). Findings: In this paper the Lucas Antimagic Labeling of Subdivision of star, Shadow graph of star, Splitting graph of star, Subdivision of Bistar, Shadow graph of Bistar, Splitting graph of Bistar are found. Novelty: It involves the mathematical formulation for labeling the edges of a given graph which in turn gives rise to a new type of labeling called the Lucas antimagic labeling.

Keywords Subdivision graph Shadow graph Splitting graph Star Bistar None
Introduction

In this paper, graph G(V,E) is considered as finite, simple and undirected with p vertices and q edges. A graph labeling is an assignment of integers to the vertices or edges. Labeled graphs are used in radar, circuit design, communication network, astronomy, cryptography etc. For detailed survey on graph labeling we refer to Gallian 1. The notion of Antimagic labeling was introduced by N.Hartsfield and G.Ringel in the year 1990. Odd antimagic labeling was introduced by V.Vilfred and L.M.Florida. Here we introduce a new notion called Lucas antimagic labeling.

A (p,q) graph G is said to be a Lucas antimagic graph if there exists a bijection f: E(G){L1,L2,Lq} such that the induced injective function f* : V(G){1,2,Lq} given by f*u=eEufe are all distinct (where E(u) is the set of edges incident to u).

Methodology

Definition 2.1: Lucas number is defined by

\mathrm{L}_{\mathrm{n}}= \begin{cases}2 & \text { if } \mathrm{n}=1 \\ 1 & \text { if } \mathrm{n}=2 \\ L_{n-1}+L_{n-2} & \text { if } \mathrm{n}>2\end{cases}

The first few Lucas numbers are 2,1,3,4,7,11,18,29,47,…﻿

Definition 2.2: A (p,q) graph G is said to be a Lucas antimagic graph if there exists a bijection f:\;E(G)\rightarrow\{L_1,L_2,\cdots L_q\} such that the induced injective function f^\ast\;:\;V(G)\rightarrow\{1,2,\dots{\sum{L_q}}\} given by f^\ast\left(u\right)={\sum_{e\in E\left(u\right)}{f\left(e\right)}}\;are all distinct (where E(u) is the set of edges incident to u).

Definition 2.3: 2 The Subdivision graph is acquired from the graph G by including a new vertex between each pair of adjacent vertices of the graph G and it is denoted by S(G).

Definition 2.4: 3 The Shadow graph D_2(G) of a connected graph G is formed by taking two copies of G say G^' and G^{''}. Join each vertex u^'in G^' to the neighbours of the corresponding vertex v^' in G^{''}.

Definition 2.5: 4 The Splitting graph S^'(G) of a graph G is acquired by including a new vertex v^'corresponding to each vertex v of G such that N(v)\;=\;N(v^').

Results and Discussion

Theorem 3.1:

S(K_{1,n})(n\geq2) where S(G) denotes subdivision of G is Lucas antimagic graph.

Proof:

Let V(S\left(K_{1,n}\right))=\{u,u_i,v_i:1\leq i\leq n\}

E(S\left(K_{1,n}\right))=\{uu_i,u_iv_i:1\leq i\leq n\}

Define a function f:\;E(S(K_{1,n}))\rightarrow\{L_1,L_2,\dots L_q\} by

f\left(uu_i\right)=L_{n+i}\;,1\leq i\leq n\;,\;\;\;\;\;\;\;f(u_iv_i)=\;L_{n+1-i}\;,\;1\leq i\leq n

The induced function

f^\ast\;:\;V(S(K_{1,n}))\rightarrow\{1,2,\dots\sum L_q\} is given by

f^\ast(u)\;=\sum\nolimits_{i=1}^nL_{n+i}

f^\ast\left(u_{i\;}\right)=L_{n+1-i}+L_{n+i} , 1\leq i\leq n

f^\ast\left(v_{i\;}\right)=L_{n+1-i} , 1\leq i\leq n

We observe that the vertices are all distinct.

Hence S(K_{1,n}) is Lucas antimagic graph.

Example 3.2: Subdivision graph S(K_{1,5}) and its Lucas antimagic labeling.

Theorem 3.3:

The shadow graph D_2(K_{1,n})(n\geq2) is Lucas antimagic Graph.

Proof:

Let V(D_2(K_{1,n}))\;=\;\{u,v,u_i,v_i\;\;:1\leq i\leq n\;\}

E(D_2(K_{1,n}))\;=\;\{uu_i\;,vv_i\;,vu_i\;,uv_i\;:\;\;1\leq i\leq n\;\}

Define a function f:\;E(D_2(K_{1,n}))\rightarrow\{L_1,L_2,\dots L_q\} by

f\left(uu_i\right)=L_i\;,1\leq i\leq n\;,\;\;\;\;\;\;\;

f(vv_i)=\;L_{n+i}\;,\;1\leq i\leq n

f(vu_i)=\;L_{2n+i}\;,\;1\leq i\leq n

f(uv_i)=\;L_{3n+i}\;,\;1\leq i\leq n

The induced function f^\ast\;:\;V(D_2(K_{1,n}))\rightarrow\{1,2,\dots\sum L_q\} is given by

f^\ast\left(u\right)=\sum\nolimits_{i=1}^nL_i+\sum\nolimits_{i=1}^nL_{3n+i}

f^\ast\left(v\right)=\sum\nolimits_{i=1}^nL_{n+i}+\sum\nolimits_{i=1}^nL_{2n+i}

f^\ast\left(u_{i\;}\right)=L_i+L_{2n+i}\;\;,1\leq i\leq n

f^\ast\left(v_{i\;}\right)=L_{n+i}+L_{3n+i}\;\;,1\leq i\leq n

We observe that the vertices are all distinct.

Hence D_2(K_{1,n}) is Lucas antimagic graph.

Example 3.4: Shadow graph D2(K1,3) and its Lucas antimagic labeling.

Theorem 3.5:

The splitting graph S^'(K_{1,n})(n\geq2) is Lucas antimagic Graph.

Proof:

Let V(S^'(K_{1,n}))\;=\;\{u,v,u_i,v_i\;\;:1\leq i\leq n\;\}

E(S^'(K_{1,n}))\;=\;\{uu_i,\;vv_i,\;uv_i\;:\;\;1\leq i\leq n\;\}

Define a function f:\;E(S^'(K_{1,n}))\rightarrow\{L_1,L_2,\dots L_q\} by

f\left(uu_i\right)=L_i\;,1\leq i\leq n\;,

f(uv_i)=\;L_{2n+i}\;,\;1\leq i\leq n

f(vv_i)=\;L_{2n+1-i}\;,\;1\leq i\leq n

The induced function f^\ast\;:\;V(S^'(K_{1,n}))\rightarrow\{1,2,\dots\sum L_q\} is given by

f^\ast\left(u\right)=\sum\nolimits_{i=1}^nL_i+\sum\nolimits_{i=1}^nL_{2n+i}

f^\ast\left(v\right)=\sum\nolimits_{i=1}^nL_{2n+1-i}

f^\ast\left(u_{i\;}\right)=L_i\;\;,1\leq i\leq n

f^\ast\left(v_{i\;}\right)=L_{2n+1-i}+L_{2n+i}\;\;,1\leq i\leq n

We observe that the vertices are all distinct.

Hence S^'(K_{1,n}) is Lucas antimagic graph.

Example 3.6:The Splitting graph S^'(K_{1,3}) and its Lucas antimagic labeling.

Theorem 3.7:

S(B(m,n))(m\geq2,n\geq2) where S(G) denotes subdivision of G is Lucas antimagic graph.

Proof:

Let V(S\left(B\left(m,n\right)\right))=\{u,v,w,u_i,u_i^':1\leq i\leq m\;,w_j,w_j^':1\leq j\leq n\}

E(S\left(B\left(m,n\right)\right))=\{uv,vw,uu_i,u_iu_i^':1\leq i\leq m\;,ww_j,w_jw_j^':1\leq j\leq n\}

Define a function f:\;E(S(B\left(m,n\right)))\rightarrow\{L_1,L_2,\dots L_q\} by

f\left(uv\right)=L_1,\;\;\; f\left(vw\right)=L_2

f\left(uu_i\right)=L_{2+i}\;,1\leq i\leq m\;,\;\;\;\;\;\;\;f(u_iu_i^')=\;L_{m+2+i}\;,\;1\leq i\leq m

f\left(ww_j\right)=L_{2m+2+j}\;,1\leq j\leq n\;,\;\;\;\;\;\;\;f(w_jw_j^')=\;L_{2m+n+2+j}\;,\;1\leq j\leq n

For each edge label f, the induced vertex label f^\astis defined by

f^\ast\left(u\right)=L_1+\sum\nolimits_{i=1}^mL_{2+i}

f^\ast\left(v\right)=L_1+L_2

f^\ast\left(w\right)=L_2+\sum\nolimits_{j=1}^nL_{2m+2+j}

f^\ast\left(u_{i\;}\right)=L_{2+i}+L_{m+2+i} , 1\leq i\leq m

f^\ast({u_i}^')=L_{m+2+i} , 1\leq i\leq m

f^\ast\left(w_{j\;}\right)=L_{2m+2+j}+L_{2m+n+2+j} , 1\leq j\leq n

f^\ast({w_j}^')=L_{2m+n+2+j} , 1\leq j\leq n

We observe that the vertices are all distinct.

Hence S(B\left(m,n\right)) is Lucas antimagic graph.

Example 3.8: Subdivision graph \;S(B\left(2,3\right)) and its Lucas antimagic labeling.

Theorem 3.9:

The Splitting graph S^'(B(m,n))(m\geq2,n\geq2) is Lucas antimagic graph.

Proof:

Let V(S^'\left(B\left(m,n\right)\right))=\{u_0,{u_0}^',u_i,{u_i}^':1\leq i\leq m\;,v_0,{v_0}^',v_j,{v_j}^':1\leq j\leq n\}

E(S^'\left(B\left(m,n\right)\right))=\{u_0{u_i}^',u_0u_i,{u_0}^'u_i\;:1\leq i\leq m\;,

v_0v_j^',v_0v_j,v_0^'v_j:1\leq j\leq n\;,u_0v_0,u_0v_0^',u_0^'v_0\}

Define a function f:\;E(S^'(B\left(m,n\right)))\rightarrow\{L_1,L_2,\dots L_q\} by

f\left(u_0v_0\right)=L_1,\; f\left(u_0{v_0}^'\right)=L_2 , f\left({u_0}^'v_0\right)=L_3

f\left(u_0u_i^'\right)=L_{3+i}\;,1\leq i\leq m\;,\;\;\;\;\;\;\;f(u_0u_i)=\;L_{m+3+i}\;,\;1\leq i\leq m

f(u_0^'u_i)=\;L_{2m+3+i}\;,\;1\leq i\leq m

f\left(v_0v_j^'\right)=L_{3m+3+j}\;,1\leq j\leq n\;,\;\;\;\;\;\;\;f(v_0v_j)=\;L_{3m+n+3+j}\;,\;1\leq j\leq n

f(v_0^'v_j)=\;L_{3m+2n+3+j}\;,\;1\leq j\leq n

For each edge label f, the induced vertex label f^\astis defined by

f^\ast\left(u_0\right)=L_1+L_2+\sum\nolimits_{i=1}^mL_{3+i}+\sum\nolimits_{i=1}^mL_{m+3+i}

f^\ast\left(u_0^'\right)=\sum\nolimits_{i=1}^mL_{2m+3+i}+L_3

f^\ast\left(u_i\right)=L_{m+3+i}+L_{2m+3+i\;},1\leq i\leq m

f^\ast\left({{u_i}^'}_{\;}\right)=L_{3+i}\;,1\leq i\leq m

f^\ast\left(v_0\right)={{\sum\nolimits_{j=1}^n{L_{3m+3+j}}}+{\sum\nolimits_{j=1}^n{L_{3m+n+3+j\;\;}}}+L}_1+L_3

f^\ast\left({v_0}^'\right)=L_2+{\sum\nolimits_{j=1}^n{L_{3m+2n+3+j}}}

f^\ast\left(v_j\right)=L_{3m+n+3+j}+L_{3m+2n+3+j}\;,1\leq j\leq n

f^\ast\left({v_j}^'\right)=L_{3m+3+j\;\;},1\leq j\leq n

We observe that the vertices are all distinct.

Hence S^'(B\left(m,n\right)) is Lucas antimagic graph.

Example 3.10: The Splitting graph S^'(B(2,3)) and its Lucas antimagic labeling.

Theorem 3.11:

The Shadow graph D_2(B\left(m,n\right))(m\geq2,n\geq2) is Lucas antimagic graph.

Proof:

Let V(D_2\left(B\left(m,n\right))\right)=\{{u_0}^',{u_0}^{''},{u_i}^',{u_i}^{''}:1\leq i\leq m\;,{v_0}^',{v_0}^{''},{v_j}^',{v_j}^{''}:1\leq j\leq n\}

E(D_2(B\left(m,n\right)))=\{{u_0}^'{u_i}^',{u_0}^{''}{u_i}^',{u_0}^{''}{u_i}^{''},{u_0}^'{u_i}^{''}:\;\;\;1\leq i\leq m\;,

v_0^'v_j^',v_0^{''}v_j^',v_0^{''}v_j^{''},v_0^'v_j^{''}:1\leq j\leq\;n\;,u_0^'v_0^',u_0^{''}v_0^{''}\}

Define a function f:\;E(D_2(B(m,n)))\rightarrow\{L_1,L_2,\dots L_q\} by

f\left({u_0}^'{v_0}^'\right)=L_1 , f\left({u_0}^{''}{v_0}^{''}\right)=L_2

f\left(u_0^'u_i^'\right)=L_{2+i}\;,1\leq i\leq m,

f\left(u_0^{''}u_i^'\right)=L_{m+2+i}\;,1\leq i\leq m\;,\;\;\;\;\;\;\;f(u_0^{''}u_i^{''})=\;L_{2m+2+i}\;,\;1\leq i\leq m

f(u_0^'u_i^{''})=\;L_{3m+2+i}\;,\;1\leq i\leq m

f\left(v_0^'v_j^'\right)=L_{4m+2+j}\;,1\leq j\leq n\;,\;\;\;\;\;\;\;f(v_0^{''}v_j^')=\;L_{4m+n+2+j}\;,\;1\leq j\leq n

f(v_0^{''}v_j^{''})=\;L_{4m+2n+2+j}\;,\;1\leq j\leq n

f(v_0^'v_j^{''})=\;L_{4m+3n+2+j}\;,\;1\leq j\leq n

For each edge label f, the induced vertex label f^\astis defined by

f^\ast\left(u_0^'\right)=L_1+\sum\nolimits_{i=1}^mL_{2+i}+\sum\nolimits_{i=1}^mL_{3m+2+i}

f^\ast\left(u_i^'\right)=L_{2+i}+L_{m+2+i\;\;\;},\;1\leq i\leq m

f^\ast\left({u_i}^{''}\right)=L_{2m+2+i}+L_{3m+2+i\;},1\leq i\leq m

f^\ast\left({u_0}^{''}\right)={\sum\nolimits_{i=1}^m{L_{2m+2+i}}}+{\sum\nolimits_{i=1}^m{L_{m+2+i}}}+L_2\;

f^\ast\left(v_0^'\right)=\sum\nolimits_{j=1}^nL_{4m+2+j}+\sum\nolimits_{j=1}^nL_{4m+3n+2+j}+L_1\;

f^\ast\left({v_j}^'\right)=L_{4m+2+j\;}+L_{4m+n+2+j\;\;},1\leq j\leq n

f^\ast\left(v_j^{''}\right)=L_{4m+2n+2+j\;\;}+L_{4m+3n+2+j\;\;},1\leq j\leq n

f^\ast\left(v_0^{''}\right)=\sum\nolimits_{j=1}^nL_{4m+2n+2+j}+\sum\nolimits_{j=1}^nL_{4m+n+2+j}+L_2

We observe that the vertices are all distinct.

Hence D_2(B\left(m,n\right)) is Lucas antimagic graph.

Example 3.12: The Shadow graph D_2(B\left(2,2\right)) is Lucas antimagic graph.

Conclusion

In this study, the concept of Lucas antimagic labeling is introduced and it is proved that various star related graphs are Lucas antimagic. Similar investigations are in process.

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